ERGODIC THEOREMS FOR NONCONVENTIONAL ARRAYS AND AN EXTENSION OF THE SZEMEREDI THEOREM

作者：Kifer Yuri^*

来源：Discrete and Continuous Dynamical Systems, 2018, 38(6): 2687-2716.

DOI：10.3934/dcds.2018113

• Szemeredi theorem
• multiple recurrence
• nonconventional averages
• triangular arrays

The paper is primarily concerned with the asymptotic behavior as N ->infinity of averages of nonconventional arrays having the form N-1 Sigma(N)(n=1), Pi T-l(j=1)Pj(n,N) f(j) where f(j)'s are bounded measurable functions, T is an invertible measure preserving transformation and P-j's are polynomials of n and N taking on integer values on integers. It turns out that when T is weakly mixing and P-i(n, N) = p(j)n + q(j)N are linear or, more generally, have the form P-j (n, N) = P-j(n) + Q(j)(N) for some integer valued polynomials P-j and Q(j) then the above averages converge in L-2 but for general polynomials P-j of both n and N the L-2 convergence can be ensured even in the "conventional" case l = 1 only when T is strongly mixing while for l > 1 strong 2l-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemeredi's theorem saying that for any subset of integers Lambda with positive upper density there exists a subset N-Lambda of positive integers having uniformly bounded gaps such that for N is an element of N-Lambda and at least epsilon N, epsilon > 0 of n's all numbers p(j)n + q(j)N , j = 1, ..., l, belong to Lambda. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemeredi theorem.

• 出版日期2018-6